Graphics Reference
In-Depth Information
(
b
a
)
2!
f
(
b
)
f
(
a
)
(
b
a
)
f
(
a
)
f
(
a
)
(
b
a
)
(
n
1)!
f
(
a
)
(
b
c
)
(
b
a
)
(
n
1)!
...
f
(
c
).
for some
c
, with
a
c
b
.
Maclaurin's Theorem
qn 37
If
f
: [0,
x
]
R
is differentiable
n
times, then
x
2!
f
(0)
...
x
(
n
1)!
f
(0)
x
n
!
f
(
x
),
f
(
x
)
f
(0)
xf
(0)
for some
, with 0
1.
Definition
The power series
h
2!
f
(
a
)
...
h
n
!
f
(
a
)
...
f
(
a
h
)
f
(
a
)
hf
(
a
)
is called the Taylor series of
f
at
a
.
Definition
The power series
x
2!
f
x
n
!
f
f
(
x
)
f
(0)
xf
(0)
(0)
...
(0)
...
is called the Maclaurin series for
f
.
Theorem
qn 38
The Taylor series expansion of a function at
a
converges to
f
(
a
h
) if and only if the difference
between the
n
th partial sum of the series and
f
(
a
h
) is a null sequence as
n
; or, in other
words, if the remainders form a null sequence.
Historical note
Early in the seventeenth century Cavalieri aMrmed that there was
always a tangent parallel to a chord, and this must count as an
embryonic form of the Mean Value Theorem.
In 1691, Michel Rolle proved that between two adjacent roots of a
polynomial
f
(
x
) there was a root of
f
(
x
). His definition of
f
(
x
) was
algebraic (so that, if
f
(
x
)
x
, then
f
(
x
)
nx
) and his proof was like
an application of the Intermediate Value Theorem in that he claimed
that
f
had a different sign at adjacent roots of
f
and therefore must be
zero somewhere between the roots.
